8r^2+14r-84=0

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Solution for 8r^2+14r-84=0 equation: 


Simplifying
8r2 + 14r + -84 = 0

Reorder the terms:
-84 + 14r + 8r2 = 0

Solving
-84 + 14r + 8r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + 7r + 4r2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-42 + 7r + 4r2)' equal to zero and attempt to solve:

Simplifying
-42 + 7r + 4r2 = 0

Solving
-42 + 7r + 4r2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-10.5 + 1.75r + r2 = 0

Move the constant term to the right:

Add '10.5' to each side of the equation.
-10.5 + 1.75r + 10.5 + r2 = 0 + 10.5

Reorder the terms:
-10.5 + 10.5 + 1.75r + r2 = 0 + 10.5

Combine like terms: -10.5 + 10.5 = 0.0
0.0 + 1.75r + r2 = 0 + 10.5
1.75r + r2 = 0 + 10.5

Combine like terms: 0 + 10.5 = 10.5
1.75r + r2 = 10.5

The r term is 1.75r.  Take half its coefficient (0.875).
Square it (0.765625) and add it to both sides.

Add '0.765625' to each side of the equation.
1.75r + 0.765625 + r2 = 10.5 + 0.765625

Reorder the terms:
0.765625 + 1.75r + r2 = 10.5 + 0.765625

Combine like terms: 10.5 + 0.765625 = 11.265625
0.765625 + 1.75r + r2 = 11.265625

Factor a perfect square on the left side:
(r + 0.875)(r + 0.875) = 11.265625

Calculate the square root of the right side: 3.356430396

Break this problem into two subproblems by setting 
(r + 0.875) equal to 3.356430396 and -3.356430396.


Subproblem 1
r + 0.875 = 3.356430396

Simplifying
r + 0.875 = 3.356430396

Reorder the terms:
0.875 + r = 3.356430396

Solving
0.875 + r = 3.356430396

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + r = 3.356430396 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + r = 3.356430396 + -0.875
r = 3.356430396 + -0.875

Combine like terms: 3.356430396 + -0.875 = 2.481430396
r = 2.481430396

Simplifying
r = 2.481430396


Subproblem 2
r + 0.875 = -3.356430396

Simplifying
r + 0.875 = -3.356430396

Reorder the terms:
0.875 + r = -3.356430396

Solving
0.875 + r = -3.356430396

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.875' to each side of the equation.
0.875 + -0.875 + r = -3.356430396 + -0.875

Combine like terms: 0.875 + -0.875 = 0.000
0.000 + r = -3.356430396 + -0.875
r = -3.356430396 + -0.875

Combine like terms: -3.356430396 + -0.875 = -4.231430396
r = -4.231430396

Simplifying
r = -4.231430396


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {2.481430396, -4.231430396}
Solution
r = {2.481430396, -4.231430396}

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